9 Unbiasedness of OLS Under TS.1 through TS.3, OLS estimators are Unbiased conditional on X, and therefore unconditionally E(u t X) = 0, t = 1, 2,..., n is a very strong assumption, often not verified Suppose: CrimeRate t = β 0 + β 1 PolicePerCapita t + u t in a given city u would need to be uncorrelated with current, past and future values of PolicePerCapita. We can accept u is uncorrelated with current and past values of the regressor. But clearly, an increase in u today is likely to lead politicians to increase PolicePerCapita in the future! TS.3 fails! Suppose: FarmYield t = β 0 + β 1 Labor t + β 2 Rainfall t + u t for a given farm u would need to be uncorrelated with current, past and future values of Labor. But maybe if last year s u was low (some plague) the farmer will not be able to hire as many workers next year. Ok with Rainfall, it will most likely not affect u João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

10 Unbiasedness of OLS (Cont.) We do not worry if u is correlated with past regressors because we can easily solve this problem: just include past regressors, use a distributed lag model But we cannot have u influencing in any way future regressors! (at least to guarantee unbiasedness) Omitted variable bias can be analyzed in the same way as for a cross-section An alternative assumption, closer to the cross-sectional case is: E(u t x t ) = 0. We would say the x s are contemporaneously exogenous. Contemporaneous exogeneity will only be sufficient in large samples João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

12 Variance of OLS Estimators (Cont.) With TS.1 through TS.5, the OLS variances in the time-series case are the same as in the cross-section case: Var(ˆβ X) = σ 2 (X X) 1 Var( ˆβ j ) = σ 2 SST j (1 R 2 j ) The estimator of σ 2 is also the same and remains unbiased OLS remains BLUE With the additional assumption of normal errors, inference is the same Assumption TS.6 (Normality) The errors are independent of X and are independent and identically distributed as Normal(0,σ 2 ) João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

13 Time Series with Trends Economic time series often have a trend If two series are trending together, we can t assume that the relation is causal We must always control for unobserved factors that can cause the trends. Otherwise we have a spurious regression problem João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

15 Adding trends in a regression We should add a trend (usually linear) to the model if either the dependent variable or the independent variables are trending y t = β 0 + β 1 x t1 + β 2 x t β k t + u t If Assumptions TS.1 to TS.3 hold in this model, leaving the trend out would in general lead to biased estimates of the remaining parameters, specially if the other regressors are trending Adding a linear trend term to a regression is the same thing as using detrended series in a regression Detrending a series involves regressing each variable in the model on t and a constant. The residuals form the detrended series Then perform the regression with detrended variables (don t need intercept, it will equal 0). It will give exactly the same estimates as the regression above João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

16 R 2 with trending data Time-series regressions with trends tend to have a very high R 2 Should therefore look at the R 2 from the regression with detrended data This R 2 better reflects how well the x t s explain y t Can also use an adjusted R 2 from the regression with detrended data João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

17 Beer consumption Seasonality Often time-series data exhibits seasonal behavior Seasonality should be corrected by, e.g., regressing each of the seasonal variables on a set of seasonal dummies Can seasonally adjust before running the regression (take the residuals from the previous regression) Should look at R-squared only on adjusted data (as for trends) João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

18 Important types of Stochastic Processes A stochastic process is stationary if for every collection of time indices 1 t 1 <... < t m the joint distribution of (x t1,..., x tm ) is the same as that of (x t1 +h,...x tm+h) for h 1 Otherwise the process is said to be nonstationary Stationarity implies that the x t s are identically distributed and that the nature of any correlation between adjacent terms is the same across all periods João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

19 Covariance Stationary and Weakly Dependent Processes A stochastic process is covariance stationary if E(x t ) is constant, Var(x t ) is constant and for any t, h 1, Cov(x t, x t+h ) depends only on h and not on t A stationary time series is weakly dependent if x t and x t+h are almost independent as h increases If for a covariance stationary process Corr(x t, x t+h ) 0 as h, we say this covariance stationary process is weakly dependent Need weak dependence to use Laws of Large Numbers and Central Limit Theorems João Valle e Azevedo (FEUNL) Econometrics Lisbon, May / 21

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